Smooth coalgebra: testing vector analysis

Abstract

Processes are often viewed as coalgebras, with the structure maps specifying the state transitions. In the simplest case, the state spaces are discrete, and the structure map simply takes each state to the next states. But the coalgebraic view is also quite effective for studying processes over structured state spaces, e.g. measurable, or continuous. In the present paper we consider coalgebras over manifolds. This means that the captured processes evolve over state spaces that are not just continuous, but also locally homeomorphic to normed vector spaces, and thus carry a differential structure. Both dynamical systems and differential forms arise as coalgebras over such state spaces, for two different endofunctors over manifolds. A duality induced by these two endofunctors provides a formal underpinning for the informal geometric intuitions linking differential forms and dynamical systems in the various practical applications, e.g. in physics. This joint functorial reconstruction of tangent bundles and cotangent bundles uncovers the universal properties and a high level view of these fundamental structures, which are implemented rather intricately in their standard form. The succinct coalgebraic presentation provides unexpected insights even about the situations as familiar as Newton's laws.

Monoidal Computer III: A Coalgebraic View of Computability and Complexity (Extended Abstract)

“…Coalgebra is the categorical toolkit for studying dynamics in general [45,48], and dynamics of computation in particular [25,43,50]. Coalgebras, as morphisms in the form X − → EX for an endofuctor E, provide a categorical view of automata, state machines, and processes with state update [20,42]; the other way around, all coalgebras can be thought of as processes with state update. In the framework on this paper, only a very special class of coalgebras will be considered, as the morphisms in the form X ×A − → X ×B, corresponding to what is usually called Mealy machines [8, 14, 17, .…”

“…This state update provides an abstract view of dynamics. Continuous dynamics can be captured in varying the same approach [42,45]. This step from X-indexed functions to X-state machines is displayed in the first row of Table 1.…”